L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s + 3·11-s − 12-s − 13-s + 3·15-s + 16-s + 3·17-s − 18-s + 7·19-s − 3·20-s − 3·22-s + 9·23-s + 24-s + 4·25-s + 26-s − 27-s − 9·29-s − 3·30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.60·19-s − 0.670·20-s − 0.639·22-s + 1.87·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s − 1.67·29-s − 0.547·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8044075456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8044075456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.411569209342716656021330590531, −7.69075319257313217209616388269, −7.13883725043338503928490257016, −6.61269462051204611677303984036, −5.41869380484404748760197356496, −4.85205263036359491825491625730, −3.61472830299585854351549324200, −3.24240102492781181261647815821, −1.56205747386749983691742213382, −0.63193682938149055791409282916,
0.63193682938149055791409282916, 1.56205747386749983691742213382, 3.24240102492781181261647815821, 3.61472830299585854351549324200, 4.85205263036359491825491625730, 5.41869380484404748760197356496, 6.61269462051204611677303984036, 7.13883725043338503928490257016, 7.69075319257313217209616388269, 8.411569209342716656021330590531